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Full text of "Treatise On Analysis Vol-Ii"


(b)    Show that the measures /xi, /-ti are equivalent, by proving that the relation
Mi(N) = 0 is equivalent to 7"(<pN) = 0.

(c)    If F1} F; are the orthogonal supplements of E3, Ei, respectively, in E, show that
F! and Fi have the same dimension (finite or infinite). (Assume that dim(Fi) is finite;
then for n ^> 2 the measures /u,n are carried by a. finite subset M of K and are zero for all
large n. Also, if vi is the restriction of ^ to M, then /zn (n ^ 2) are measures with base
vi. Deduce that T(<pM)(E) is of finite dimension and that the restriction of T to the
orthogonal supplement T(l  9?M)(E) of J(9M)(E) is topologically cyclic. Use (a) to
deduce that the measures \jJn, for n ^ 2, are concentrated on M, and hence that

7X1 - <pM)(E) = 7*(1 - ^(Ei) = T(l - <pM)(Ei).

Also, by virtue of (b), the restriction of /xi to M must be equivalent to /*i; hence
7X9>M)(Ei) and 7X<pM)(Ei) have the same finite dimension, and the result follows.)

(d)    Use (c) to show that there exists a unitary transformation C/i of E such that
^(EO = Ei.

(e)    Prove by induction that there exists a sequence (Un) of unitary transformations of
E such that Un(Ek) = E'k for 1 f k <> n and Un +1 agrees with Utt on Ek for 1 <; A; <J .
Show that this sequence tends strongly (Section 12.15, Problem 8) to a unitary trans-
formation U such that U(En)  E for all . Show also that ^ is equivalent to pn
for all n.

6.   With the notation of (15.10.9), let Vbe an operator belonging to -S?(E) which commutes
with T(u) for all u e ^C(K).

(a)    Show that each of the subspaces G* is stable under V, and that V\ G* commutes
with all the operators T(<pMk u).

(b)    Up to equivalence, each of the Hifc (1 ^i^ k) can be identified with LC(^*) and
the restriction Tik(u) of T(u) to H/fc can be identified with multiplication M(u) by the
class of w, where u is any bounded ^-measurable function. Every continuous operator
V on Gfc may be written in the form of a matrix

operating on the vectors x  fc)i< t^k e(Lc(^))'c considered as column matrices,
the Vu being continuous operators on LcO"*). With this notation, the restriction of
T(u) to OK is identified with the diagonal matrix all of whose diagonal elements are
equal to M(u).

Deduce that V commutes with the restrictions of all the T(u) to G* if and only if

Vv = M(vtJ)

with vtj e rc(K).

(c) Deduce that every subspace F of E which is stable with respect to Tis the Hilbert
sum of subspaces F* c G* stable under T, and that each Ffc may be described as follows.
Consider a partition of Mk consisting of k ^-measurable sets Lik (1 ^ / ^ k)t and an
invertible matrix (w^) with ^ rows and k columns, whose entries wu belong to^ 2, show that there would exist a sequence (gm)